ALTERNATE COMPUTER MODELS OF FIRE CONVECTION PHENOMENA FOR THE HARVARD COMPUTER FIRE CODE. by Douglas K. Beller

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1 ALTERNATE OMPUTER MODELS OF FIRE ONVETION PHENOMENA FOR THE HARVARD OMPUTER FIRE ODE by Douglas K. Beller A Thesis Submitted to the Faculty of the WORESTER POLYTEHNI INSTITUTE in partial fulfillment of the requirements for the Degree of Master of Science in Fire Protection Engineering by Douglas K Beller August 1987 APPROVED: Dr. raig L. Beyler, major Advisor Prof. David A. Lucht, Director, enter for Firesafety Studies

2 ABSTRAT Alternate models for extended ceiling convection heat transfer and ceiling vent mass flow for use in the Harvard omputer fire ode are developed. These models differ from current subroutines in that they explicitly consider the ceiling jet resulting from the fire plume of a burning object. The Harvard omputer fire ode (F) was used to compare the alternate models against the models currently used in F at Worcester Polytechnic Institute and with other available data. The results indicate that convection heat transfer to the ceiling of the enclosure containing the fire may have been previously underestimated at times early in the fire. Also, the results of the ceiling vent model provide new insight into ceiling vent phenomena and how ceiling vents can be modeled given sufficient experimental data. this effort serves as a qualitative verification of the models as implemented; complete quantitative verification requires further experimentation. Recommendations are also included so that these alternate models may be enhanced further. ii

3 AKNOWLEDGEMENTS I would like to thank my advisors: Prof. W. H. Kistler for providing guidance concerning some of the more rigorous mathematics, Prof. B. J. Savilonis for helping me improve my "intuitive feel" for the phenomena being modeled, Prof. J. R. Barnett for helping me overcome program debugging problems and for useful information regarding F, and, my major advisor, Prof.. L. Beyler for having that extra bit of knowledge and insight, into both the phenomena and the modeling, that added to the quality of this thesis. I would also like to acknowledge the support of Gary Hartley and Sal Vella of General Dynamics' Electric Boat Division: without their support this project would have been that much more difficult. Thanks are also extended to J. Muldoon, WPI computer center nighttime operator, for being that rare operator who is actually helpful. And a very special thanks to Joanne for putting up with me for the duration of it all. which... This work is dedicated to my parents, without whom iii

4 TABLE OF ONTENTS HAPTER PAGE Abstract Acknowledgements Table of ontents List of Figures List of Tables ii iii iv viii xi 1.0) Introduction 1 1.1) Discussion 1 1.2) Purpose 5 2.0) Extended eiling onvection Heat Transfer 7 2.1) urrent F Model 7 2.2) Available Models 9 2.3) Alternate Model ) Assumptions ) eiling Proper (A) Heat Transfer oefficient (B) Gas Temperature () Surface Temperature ) Heated Wall (A) Heat Transfer oefficient (B) Gas Temperature () Surface Temperature 29 iv

5 TABLE OF ONTENTS (cont.) HAPTER PAGE 2.3-4) Model Formulation (A) Point Selection Process (B) Averaging alculations ) Required Programming ) NVW ) VENTA ) VNTNT ) BIGHRR ) QSTAR ) FTRS ) TMPA ) FROVRH ) HTX ) RE ) NVL ) TMPW ) Model Verification ) eiling Vent Mass flow Rate ) urrent F Model ) Available Models ) Alternate Models 110 v

6 TABLE OF ONTENTS (cont.) HAPTER PAGE 3.3-1) Assumptions ) Small Fires ) Large Fires (A) Vent Near Plume Axis (B) Vent Far From Plume Axis ) Required Programming ) VMF ) VENTA ) VA ) VB ) VNTNT ) BIGHRR ) VB ) VB ) QSTAR ) FTRS ) TMPA ) FROVRH ) UF ) UV ) Model Verification vi

7 TABLE OF ONTENTS (cont.) HAPTER PAGE 4.0) omparison of Results for the Standard ase ) EHTX ) VMFR ) EHTX and VMFR Together ) Summary and Discussion of Results ) EHTX ) VMFR ) omputational onsiderations ) Recommendations ) EHTX ) Further Model Verification ) Future Modeling onsiderations ) VMFR ) User's Guide ) Input ) Output ) Program Messages onclusion 185 References 187 Appendix A: urrent F/WPI eiling Vent Model 190 Appendix B: Listings of Required Subroutines 194 vii

8 LIST OF FIGURES FIGURE DESRIPTION PAGE 1.1(a) Fire Transport Phenomena: Room w/ Open Door 3 1.1(B) Fire Transport Phenomena: nearly losed room (A) Heat Transfer to an Unconfined eiling (B) Heat Transfer to a confined eiling (A) Early Time eiling Jet-Wall Interaction (B) eiling jet-wall Interaction with Upper layer (A)1 Fourth Derivative of q(r/h) vs r/h (A)2 Absolute Value of Fourth Derivative (A)3 (1/Absolute Value of Fourth Derivative)** (A)4 Variable mesh Generator Algorithm Flowchart (A)5 qc" = q(r/h) = h * Tg (A)6 eiling Representation for Standard ase (A) Alternate EHTX Model Flowchart (A) The Unconfined eiling Scenario (B) Typical Radial Surface Temp Distribution () Imp Point Temp Rise: Sm Rm, Sm Stdy Fire (D) Imp Pint Temp Rise: Sm Rm, Lrg Stdy Fire (E) Imp Point Temp Rise: Lrg Rm, Sm Stdy Fire (F) Imp Point Temp Rise: Lrg Rm, Lrg Stdy Fire (G) Imp Point Temp Rise: Sm Rm, T**2 Fire (H) Imp Point Temp Rise: Lrg Rm, T**2 Fire 87 viii

9 LIST OF FIGURES (cont.) FIGURE DESRIPTION PAGE 2.5(I) Ref. 6 Normalized Impingement Point Heat Flux (J) Alt EHTX Model Normalized eiling Heat Flux (A) urrent Model: eiling Vent Scenario (B) Thomas, et al, Model: eiling Vent Scenario () eiling jet Rise Through a eiling Vent (A) Alternate VMFR Model Flowchart (B) Alternate VMFR Model Decision Flowchart (A) Upper Gas Temperatures (B) eiling Surface Temperatures () Layer to Wall onvective Heat Flow (A) eiling Vent flow - Small Vent (B) Pressure at Floor - Small Vent () Upper Door flow - Small Vent (D) Lower door Flow - Small Vent (E) eiling Vent Flow - Large Vent (F) Pressure at Floor - Large Vent (G) Upper Door Flow - Large Vent (H) Lower door Flow - Large Vent (I) Upper Layer Temperature - Large Vent (J) Upper Layer Depth - Large Vent 148 ix

10 LIST OF FIGURES (cont.) FIGURE DESRIPTION PAGE 4.3(A) eiling Vent Flow - Both Alternate Models (B) Pressure at Floor - Both Alternate Models () Upper Door Flow - Both Alternate Models (D) Lower door Flow - Both Alternate Models (E) Upper Layer Temp - Both Alternate Models (F) eiling Surface Temp - Both Alternate Models (G) Layer to Wall nv Heat flow - Both Alt Models (H) Upper Layer Depth - Both Alternate Models (A) Input Processing Example (A) Tabular Output Example 180 x

11 LIST OF TABLES FIGURE DESRIPTION PAGE 2.3-4(A)1 Preliminary Spacing Scheme Results (A)2 Final Spacing Scheme Results (A)3 Point Locations for eiling alculations (A) Annular Heat Transfer Area Max/Min Radii (A) eiling Material Physical Properties (B) Description of Six fire Scenarios () alculation Information: Unconfined eiling (A) alculation Information: EHTX Standard ases (A) alculation Information: VMFR Standard ases (A) alculation Information: "Both" Standard ases 162 xi

12 ALTERNATE OMPUTER MODELS FOR FIRE ONVETION PHENOMENA 1.0) INTRODUTION Historically, modeling enclosure fires with computers has fallen into two categories: field models and zone models. Field models tend to be global in nature in that they explicitly consider all regions of space within the enclosure. This is accomplished by solving the equations of motion at a large number of points representing the space inside the enclosure. The heat transfer to the enclosure walls is also calculated at a number of points through the thickness of the wall. The exact number of points both for the space of the room and the walls, which comprise the mesh or grid, are determined in part by the problem at hand, part by available computational ability and part by user input. Field models can thus provide a fairly realistic representation of enclosure fire phenomena. However, the price to be paid for this realism is greatly increased computation time, which may be undesirable from a design or production point of view. Zone models, on the other hand, are not quite so time intensive because, instead of considering individual points within the enclosure, the enclosure is modeled as if it were a conglomeration of regions. In these models, a small number of regions is required and entails a control volume approach. Typically the number of regions is on the order of ten or less. In the 1

13 past these regions have consisted of the burning object (and any other target objects), the combustion zone and plume above the burning object, a lower gas layer and an upper gas layer. However, zone models are typically referred to as two-zone models; i.e., the upper and lower gas layers are the two zones of interest. 1.1) DISUSSION The implication of a two-zone enclosure fire model is that the hot, upper and cold, lower layers are homogeneous, i.e., assumed to be well-mixed and of a uniform temperature. Both assumptions are probably more valid for the lower layer than for the upper layer, especially early in the fire (see Fig. 1.1(A), Ref. 14). The upper layer, prior to flashover, has different zones of mixing and exhibits thermal stratification (see Fig. 1.1(B), Ref. 14) as well as radial temperature variations. However, the upper layer as presently modeled does not account for the dynamics associated with the upper part of the fire plume and the resultant ceiling jet, which causes some of these nonhomogenous effects. The ceiling jet is significant for the following reasons: the convective heat transfer between the extended ceiling of the enclosure and the ceiling jet can be a significant fraction of the fire's total heat release (especially during the early stages of the fire) and because the mass flow rate out of a ceiling vent or wall 2

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16 vent with a small soffit depth may be significantly affected by the momentum of the ceiling jet. The alternative models described herein attempt to overcome these inadequacies 1.2) PURPOSE The purpose of this endeavor is to provide a description of alternate computer models characterizing extended ceiling convection heat transfer (EHTX) and ceiling vent mass flow rate (VMFR) which consider the existence of a plume and ceiling jet in their formulation. Essentially, these are two different models, describing different phenomena, which can be used alone or together and which also share several subroutines. The extended ceiling is defined to be those enclosure surfaces exposed to the hot upper layer gases. As such these alternate models represent a departure from current modeling techniques. The ceiling jet is produced by the impingement of the fire plume on the ceiling. In existing models the fire plume is assumed to be "cut off" at the interface between the upper and lower layers within the enclosure. With this assumption, certain aspects of EHTX and VMFR are not included. The alternative models discussed here are an attempt to alleviate some of the restrictions imposed by cutting off the fire plume at the layer interface. The alternate models described herein have been 5

17 incorporated into the Worcester Polytechnic Institute (WPI) version of the omputer Fire ode (F). This program was originally developed at Harvard University, and described in Ref

18 2.0) EXTENDED EILING ONVETION HEAT TRANSFER In order to fully appreciate the differences between the alternate model and those currently employed, a brief description of the existing model, along with the assumptions upon which it is based, is presented next. This section also discusses some of the shortcomings of the existing models and suggests improvements to overcome them. Section 2.2 is a description of some of the models available which were considered as a possible basis for an alternative to the existing model. And finally, a more complete description of the alternate model which was actually implemented is discussed in Sect A description of the subroutines this model requires is included in Sect ) URRENT F MODEL The extended ceiling convection heat transfer (EHTX) model currently used by F assumes that the EHTX is a function of uniform extended ceiling surface temperature, instantaneous upper layer temperature, and a heat transfer coefficient. This heat transfer coefficient is defined as (Ref. 16): h = min[50,5 (50 + 5)(T 100 L TA ) ](W m K) In other words, the heat transfer coefficient has a minimum value of 5 W/m-K and rises linearly with temperature over 7

19 a temperature rise of 100K up to 50W/m-K; T L is the upper layer temperature and T A is the ambient temperature. These two values (5 and 50) are used as the max/min values of the heat transfer coefficient used as input by F. Therefore, these values can be user defined. In view of our latest understanding of enclosure fires, the existing model is inadequate in several ways. First of all, although not stated above, the instantaneous upper layer temperature is assumed to be uniform throughout the layer. This is not necessarily the case due to the presence of the fire plume and the resultant ceiling jet. Indeed, the ceiling jet will exhibit a radial temperature distribution (decreasing with increasing distance from the fire plume axis). Also, it is this ceiling jet which would tend to drive the convection heat transfer to the ceiling more than the uniform hot layer temperature. Furthermore, the extended ceiling surface temperature will not be uniform since it is induced by the impingement of the plume on the ceiling and the resultant ceiling jet. Specifically, the ceiling surface will exhibit a radial temperature distribution while the surface temperature distribution of the portion of the wall exposed to hot gases will be monotonically decreasing with increased distance from the ceiling. Therefore, both the adjacent surface gas temperature and the extended ceiling surface temperature are functions of distance from the plume axis and the EHTX 8

20 model should incorporate this phenomena. While it is true that the heat transfer coefficient is an indirect function of temperature, it seems likely that it is a stronger function of the velocity of the gases adjacent to the ceiling surface and the radial distance from the plume axis. The observed behavior of fire plumes impinging on ceilings has been: high radial gas velocities near the ceiling impingement point which decrease as the jet approaches the enclosure walls. In view of this, the alternate model should be based on (at least) a spatially dependent heat transfer coefficient. Optimally the EHTX model should also incorporate the phenomena detailed above and have provision for: convective heat transfer to the entire ceiling and all parts of the heated wall, upper layer effects (i.e., increased ambient temperatures), disruption of the ceiling jet (and, therefore, a change in EHTX) due to increased hot layer turbulence attributable to an open ceiling vent or other obstructions, and the (presumed) increase in EHTX due to more than one burning object. However, the state-of-the-art does not allow the formulation of such a complete model: therefore, the intent is to provide as comprehensive model as our current understanding allows. 2.2) AVAILABLE MODELS Reference 2 provides a review of available ceiling jet and ceiling heat transfer models. From that reference, the following models were considered: Alpert (Ref. 1), ooper (Ref. 5), Evans (Ref. 9), Heskestad and Delichatsios (Ref. 9

21 11), and You and Faeth (Ref. 18). Of these, References 5, 9 and 18 are for ceiling convection heat transfer, while 1 and 11 are for ceiling jets (i.e., from which an EHTX model could subsequently be developed). Alpert (Ref. 1) is concerned with the actuation of fire detectors, which may or may not be suitable for an alternate EHTX model. His model has the limitation of being unable to account for the heated portion of the wall. Therefore, Alpert's model will not be considered as an alternate EHTX model. ooper (Ref. 5 and 7) devises a method to calculate the heat transfer to the entire ceiling as well as the heated part of the walls (Ref. 8) which is based on data accumulated from the available literature. ooper also considers the effect of the hot upper layer, which results from confined ceilings, by preserving the average temperature of the plume and the mass flux across the layer interface. Evans (Ref. 9) is also interested in detector actuation. He modifies ooper's method by maintaining the plume width and gas velocity to develop the correlations for hot layer effects. These variables are less important than those considered by ooper when modeling EHTX and thus Evans' method will not be considered. 10

22 Upon initial inspection, the model of Heskestad and Delichatsios (Ref. 11) appears to be adequate: it is an experimental validation of the modeling relations for convective flow generated by power-law fires and it applies to the entire ceiling. However, it does not consider heat transfer to the heated parts of the walls nor does it apply to the general fire case. Therefore, the Ref. 11 formulation is not considered. The model of You and Faeth (Ref. 18) is also a heat transfer model which applies to the entire ceiling. However, it does not explicitly account for hot layer effects or heat transfer to the heated part of the walls. After weighing the advantages and disadvantages, ooper s model was chosen as the basis for the alternate EHTX model. It should be pointed out that the choice of ooper's model is somewhat pragmatic and not necessarily optimal in all aspects. There are three components to consider: ceiling jet flow, ceiling jet heat transfer and hot layer effects. onceivably each of these components could be provided by three different sources and then combined into the desired model. This approach was thought to be unnecessary since ooper's model already combines these three components into a single, comprehensive whole. Thus the problem of developing consistent interfaces between the individual, and possibly disparate, components is 11

23 avoided. No improvements to ooper s basic formulation were discerned. 2.3) ALTERNATE MODEL The model proposed by ooper in Refs. 5, 7, and 8 was chosen as the basis for an alternate EHTX model since it fulfills most of the requirements previously stated. The original model of Ref. 5 was revised and enhanced in Ref. 7 and 8. These second two references serve to expand the geometric applicability of the model and to incorporate heated wall heat transfer. Therefore, this model has the advantage of calculating the heat transfer to all points on the ceiling as well as to the heated walls. ooper also includes the effect of the hot upper layer which results from confined ceilings. In this context, confined is defined to mean that the enclosure is small enough such that a hot upper layer will form. Alternately the enclosure could be so large that an appreciable upper layer may not form. ooper's model is based on the review of other researcher's experiments. This is not to be construed as a shortcoming, however, since Ref. 2 states that even though ooper did not include the data of Heskestad and Delichatsios (Ref. 11), i.e., the correlation recommended in Ref. 2, his correlations "are in good agreement" with the data of Ref

24 This alternate EHTX model is broken into two parts: one for the convection heat transfer to the ceiling proper and one for the convection to the heated portion of the wall, i.e., wall area covered by the hot, upper layer. Ref. 5 and 7 will provide the basis for the former and Ref. 8 for the latter. These models are described in Sect and The next section details the assumptions required by the model ASSUMPTIONS This model requires several assumptions. Most of these assumptions are made to simplify the modeling and/or programming and all are subject to change pending available experimental data. The assumptions for the EHTX model are: 1) Model is applicable to smooth ceilings only. This assumption eliminates any ceiling jet flow changes due to open ceiling vents and ceiling obstructions. This may decrease the model's applicability. However, for the purpose of this model, if the model is in effect, then the ceiling jet is also present. 2) The flame axis is the geometrical center of horizontally burning objects and is not affected by ambient conditions, i.e., it does not migrate in the presence of wind or drafts caused by ventilation systems. 13

25 3) When more than one object is burning, only the object with the highest heat release rate is considered, i.e., contributions of smaller fires are assumed insignificant. (Note: As described in a later section, this assumption may neglect the geometry of the scenario if the object with the highest heat release rate is not the initially burning object.) If this assumption is not made, then the interaction of individual plumes would determine the ceiling jet characteristics. This interaction has not been studied to date so that no information exists as to how it should be modeled. This is a gross oversimplification and requires further investigation. 4) To simplify the programming, an equivalent vent radius is needed to locate the ceiling vent relative to the fire plume axis. 5) No definition has been supplied for the layer depth at which increased ambient temperatures should be considered to be significant, as discussed in Sect. 2.1 and 2.3-2(A). Therefore, this "significant" layer depth is assumed to be 14

26 twice the ceiling jet thickness or 0.24 times the floor to ceiling height. This is an arbitrary choice as to "significant" fraction of room height (see page 30 of Ref. 17 for this definition of ceiling jet thickness = 0.12 times the floor-to-ceiling height). Presumably the value of "significant" fraction of room height is critical to the subsequent computer output and its effect should be more extensively investigated. 6) Radial temperature gradients of the problem are assumed to be small enough so that in the ceiling is quasi-one dimensional in space, i.e., the in-depth ceiling coordinate (Ref. 6) EILING PROPER by: From Ref. 7, the ceiling convection heat flux can be estimated Eq q c = h T ad T ) ( s where T ad is the gas temperature distribution at the surface of an adiabatic ceiling established by the ceiling jet flow from the plume of a given fire with a given fire-to-ceiling distance; T s is the instantaneous lower (i.e., exposed) 15

27 surface ceiling temperature distribution; and h is the heat transfer coefficient based on the (T ad - T s ) temperature difference. All three of these parameters must be modeled in order for this model to be consistent and complete. This formulation was originally developed for confined ceiling scenarios by ooper in Ref. 5. However, the basis for that work is supplied by Ref. 4. In Ref. 4 ooper incorporated the work of several researchers investigating heated turbulent ceiling jet flows and unheated turbulent wall jets into a model describing the scenario of Fig (A). This was accomplished by modeling the fire's combustion zone as a point source of energy and by drawing equivalence between the momentum and mass fluxes of the free jet and of a buoyant plume at the position of their respective impingement with the ceiling surface (Ref. 4). The resultant model consisted of correlations for T ad and h as functions of fire parameters and a geometric parameter, r/h, where r is the radial distance from fire plume axis and H is the plume source-to-ceiling distance. Figure 2.3-2(A) depicts the scenario near the plume axis at early times in the fire or at later times for large, expansive ceilings. If vertical surfaces are sufficiently distant (i.e., expansive smooth ceiling with large r/h), then the ceiling jet loses most of its momentum far out in its trajectory. However, if the enclosure has a low aspect 16

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29 ratio, then the ceiling jet flow is blocked by the bounding vertical surfaces (i.e., walls) and forms a downward overturning wall jet flow which is eventually turned back inward and upward by its own buoyancy (Ref. 5), see Fig. 1.1-(A), 1-1(B), and 2.3-3(A). The blocked ceiling jet gases eventually redistribute themselves horizontally across the cross section of the enclosure. This tends to form a relatively quiescent, stably stratified upper layer, below the ceiling jet (Ref. 4), which also defines the layer interface. This layer interface defines the demarcation between the cooler, ambient air below and the hotter products of combustion and entrained air above. This layer interface drops with increasing time, see Fig. 1.1(A), while the average absolute temperature, T u, of the upper layer rises with time, i.e., as the fire continues to burn. Figure 2.3-2(B) depicts the near plume scenario as just discussed. Therefore, in Ref. 5 ooper reformulates the Ref. 4 correlations to account for Fig (B) scenario, i.e., to account for hot, upper layer effects (A) HEAT TRANSFER OEFFIIENT The formulation for the heat transfer coefficient used in the alternate EHTX model was originally developed in Ref. 4 for unconfined ceiling scenarios. Heat transfer 18

30 coefficient data from several researchers were curve fit using a least squares approach. The resultant formulation is a function of r/h for the heat transfer coefficient. Also, in order to incorporate the data from wall jet heat transfer measurements with that from plume driven heat transfer measurements, an equivalence between these two phenomena was required. A relationship was established between the measured properties of a wall jet and the properties of its equivalent buoyant plume at their respective points of impingement (Ref. 4) by developing an equivalent Reynolds number. This equivalent Reynolds number, Re H, differs from the traditional formulation by a constant, if Re H is cast in terms of the fire heat release rate and the plume source-to-ceiling distance as presented below. As stated previously, Ref. 5 enhanced the Ref. 4 model by including hot layer effects. In Ref. 7 ooper and Woodhouse increased the range of r/h to 2.2 for which the Ref. 5 correlations are applicable. be: The heat transfer coefficient, h, is defined by Eq. 6, Ref. 7 to Eq (A)1a -1/2-2/3 h {8.82ReH Pr [1 - ( ReH0.2)(r/H)],0 (r / H ) h ~ / Re Pr (r / H),0.2 H (r / H ) 1 where: h ~ 1 / 2 1 / 2 = ρ amb cpg H QH * 3 (r/h) (r / H) 0.2 Re Q H H * 1 / 2 3 / 2 1 = (g H QH * )/ ν 3 = (1 λ )Q /( ρ r amb c p T amb g 1 / 2 H 5 / 2 ) 19

31 ρ amb c p T amb = ambient density = ambient specific heat = " temperature ν = " kinematic viscosity Pr = Prandtl number = 0.7 g = acceleration of gravity H = plume source-to-ceiling distance r = distance from fire plume axis Q = energy release rate of the fire λ r = fraction of Q lost by radiation Note: since F does not currently calculate λ r, coding is included to calculate it as the radiant energy loss of the flames divided by the energy release rate of the fire (TEPZR / TEOZZ) and is calculated every time step. As the fire progresses, the upper layer depth increases and the phenomena governing the heat transfer becomes more complex. From Fig (B), two additional parameters are required, T u and (upper layer thickness) (Ref. 4). In Ref. 5 ooper develops the correction factors that allow the Ref. 4 formulations to be used for the scenarios shown in Fig (B). In this scenario, the temperature of the gases entrained by the plume above the interface is greater than that of the gas entrained below the interface. ooper submits "that once the depth of the upper layer becomes a significant fraction of H (room height)...the impact of the 20

32 now elevated upper layer temperatures on the temperature and mass flux of the upper portion of the plume will also be significant" (Ref. 5). However, ooper does not define "significant fraction". Thus, when assumption five of Sect holds, the ambient environment will be the hot upper layer, not that outside the room. Three correction factors are calculated to account for elevated upper layer temperatures: T u /T amb, and the two factors defined by Eq (A)2 and 2.3-2(A)3. Therefore, h is modified as per Ref. 5 as follows: Eq (A)1b h h ~ {8.82ReH Re -1/2 0.3 H -2/3 Pr [1 - ( Re H 0.2)(r/H )],0 2 / (r / H ) Pr (r / H),0.2 (r / H ) where: h ~ 1 / 3 1 / 3 = (H / H) (Q / Q) (T / T ) h ~ amb u (r/h ) 0.2 (r / H) (Eq. 18, Ref. 5) Eq (A)2 H H = H (1 ( / H))(T (1 Tu / T [ 2 / 3 Q * Zi u amb / T ) + amb ) 1] 1 5 (Eq. 21, Ref. 5) Eq (A)3 Q Q = 0.201(1 Q Zi T / T 2 / 3 * ) u amb + 1 (Eq. 22, Ref. 5) Q Zi * = Q* evaluated at interface elevation, Z I 21

33 T u = temperature of upper layer (K) = upper layer thickness Re H = (H / H) (T u 2 / 3 (Q / Q) / T ) 5 / 2 amb 1 / 3 (1 [(T + amb / T ) + (110.4 / T u / T amb )(1 / Re H ) amb )] (Eq. 20, Ref. 5) 2.3-2(B) GAS TEMPERATURE In addition to the heat transfer coefficient, ooper also curve fit small scale buoyant plume driven ceiling jet experimental data (Ref. 4) to arrive at an expression for T ad. This formulation was also enhanced in Ref. 5 and 7 as previously discussed. T ad represents the maximum temperature possible for the times of interest. T ad is determined by the characteristics of the plume immediately prior to impingement. In other words, the maximum gas temperature of the entire distribution is at the impingement point, and, therefore, all downstream temperatures are dependent on this temperature. The impingement point ceiling surface temperature is a function of the fire plume conditions just prior to impingement. The (T ad - T amb ) temperature difference is a function of r/h and is shown in Eq (B)1 and 2.3-2(B)2 The adiabatic gas temperature, T ad, is defined by Eq. 9, Ref. 7: Eq (B)1a T ad * = (9 r / H),0 (r / H) 8.39f(r / H),0.2 (r / H)

34 Tad Tamb where: Tad* = 2 / 3 T Q * amb H Q H * 2/3 = Q* evaluated at H Eq (B)2 f(r / H) = (1 r / H) (r / H) (1 r / H) + 2. (2r / H) (r / H) 2.4 (Eq. 7, Ref. 7) As in Sect (A), an increased temperature environment also affects the resultant gas (i.e., ceiling jet) temperature in a manner similar to the affect on the heat transfer coefficient. Therefore, correction factors are required when assumption five of Sect holds. These are the same correction factors discussed in Sect (A). Tad is modified as per Ref. 5 as follows: Eq (B)1b (9 r / H),0 (r / H) T ad * = 8.39f(r / H),0.2 (r / H) T ad Tamb where: Tad* = 2 / 3 T Q * amb H 0.2 5/2 Q H * = (Q /Q)(H /H) Q * (Eq. 19, Ref. 7) f (r / H ) = f(r / H) evaluated at H H 2.3-2() SURFAE TEMPERATURE The ceiling surface temperature for this model is calculated in a similar manner to what is currently done in F. The only difference is that, instead of only one bulk condition (i.e., one 23

35 surface temperature, one gas temperature, and one heat transfer coefficient), between, typically, three and ten local conditions may be used, depending on the size of the room. The process for selecting these local points is discussed in Sect (A) For further information, see subroutine TMPW01 as described in Ref HEATED WALL From Ref. 8, the heated wall convection heat flux can be estimated by: Eq : q = h(t ad T ) w where T ad is the gas temperature distribution adjacent to an adiabatic wall upon which a plane jet from an elevated temperature plume is impinging; T w is the wall temperature which "would generally vary with position from the stagnation point" (Ref. 8); and h is the heat transfer coefficient at the stagnation point of the ceiling jet where it impinges on the wall and is equivalent to h s. All three of these parameters must be modeled in order for this model to be consistent. In Ref. 8, ooper draws an analogy "between the flow dynamics and heat transfer at ceiling jet-wall impingement and at the line impingement of a wall and a two-dimensional, plane, free jet". To accomplish this, ooper developed a correlation for Nusselt number, 24

36 based on the small scale, experimental plane jet impingement data he reviewed. In order to apply this (small scale) formulation to ceiling jet-well impingement, i.e., by analogy, ooper chose the distance from the jet s virtual origin, X, and the momentum flux per unit width, M o, so that they simulate the ceiling jet flow immediately upstream of (i.e., near) the wall impingement point at r = D shown in Fig (A). The results are some "readily available estimates for the heat transfer from, and the mass, momentum, and enthalpy fluxes of the turned compartment fire ceiling jet [i.e., downward wall jet] as it begins its initial descent as a negatively buoyant flow along the compartment wall" (Ref. 8). This is shown in Fig (A) (Ref. 8). The "equivalence" method used in Ref. 8 is analogous to that used in Ref. 4 and is discussed briefly in Sect When the upper layer is a "significant fraction" (Sect (A)) of the room height, two scenarios are possible, as shown in Fig (B). The scenario shown in Fig (B)1 is not considered in this alternate EHTX model. Instead, the negative wall flow is assumed to not penetrate the layer interface, as shown in Fig (B)2. Also, the front of the negative wall flow is assumed to descend at the same rate as the layer interface. Thus, the heated wall area becomes the heat transfer area for the negative wall flow. 25

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39 2.3-3(A) HEAT TRANSFER OEFFIIENT In Ref. 8, ooper uses experimental data of heat transfer from an ambient temperature plane jet to an isothermal wall to arrive at the required expressions for the stagnation point heat transfer coefficient. A curve fit to the data resulted in a Nusselt number formulation for this heat transfer coefficient. ooper then extends this result to "estimate heat transfer rates, q, from elevated temperature jets to non-uniform temperature walls" in a manner analogous to that discussed in Sect This result is then recast in terms of enclosure fire parameters and is discussed below. The heat transfer coefficient, h s, is defined by Eq.15, Ref. 8: Eq (A)1a hs h h ~ h s = 0.89 Re 0.42 H Pr 1 (D / H) 1.02 where: D= wall-to-fire distance: Fig (A) When assumption five of Sect holds, i.e., as discussed in Sect (A), the ambient environment will be the hot upper layer, not that outside the room and the three correction factors discussed in Sect (A) are used. The heat transfer coefficient is defined (as per Ref. 5) by: 28

40 Eq (A)1b h s h h s h ~ = 0.89 Re H 0.42 Pr 1 (D / H ) (B) GAS TEMPERATURE The heat transfer coefficient described in Sect (A) requires T ad. That is to say, in order to use Eq (A)1 in Eq , the (T ad - T w ) temperature difference must be known. Thus, T ad is defined by Eq (B)1 with r = D () SURFAE TEMPERATURE As stated in Ref. 8, the wall temperature, T w "would generally vary with (vertical) position from [the] stagnation point". However, for this model the vertical surface temperature profile along a line down the wall a given radial distance from the fire axis, will be considered to be a constant. In other words, this assumption does not require multiple points, vertically spaced down the heated portion of the wall. Instead, only four points are needed: one for each wall. This assumption is made for two reasons: one, because h s is based on conditions just upstream of the stagnation point (i.e., where the ceiling and wall meet) and two, because at larger values of r/h, h, T ad, and T w do not vary significantly. In other words, the magnitudes of these variables tend to reach a fairly uniform value a given 29

41 distance outside the impingement zone, see Fig (A)5. Therefore, the error introduced by this assumption is not expected to be large. The heated wall surface temperature is calculated in a manner similar to what is currently done by F. The only difference is that, instead of only one bulk condition, four local conditions are used. These are the conditions present at the distances, D, as defined in Eq (A)1a. Therefore, a total of seven to fourteen points may be used to characterize the convective heat transfer to the entire extended ceiling. The selection of these points is described in Sect (A). 30

42 2.3-4) MODEL FORMULATION The correlations provided by ooper in Ref. 5 and 7 are intended to evaluate local conditions along the ceiling proper. The correlations of Ref. 8 are intended to evaluate the local conditions just upstream of the stagnation point of the ceiling jet before it contacts the wall. Thus, the local conditions at any point of the extended ceiling can be determined, subject to the assumptions of Sect and However, for this alternate EHTX model, the interest is in the convective heat transfer to the extended ceiling as a whole. The crux of the problem is choosing the points along the extended ceiling which, taken in combination, yield a fairly accurate estimate of the energy convected to extended ceiling (A) POINT SELETION PROESS Essentially, the problem is how to apply equations which describe local conditions i.e., those of ooper, in a global manner. The solution is to integrate the equations over the range of interest. If this can be done analytically then an accurate solution can be obtained. However, if done numerically, then the associated error must be taken into consideration, as well as which points, and how many of them, to use. First the point 31

43 selection process for the ceiling proper will be discussed and then for the heated wall. Start with the governing equations: expressions are available which describe the heat transfer coefficient (Eq (A)1) and near surface gas temperature (Eq (B)1) as functions of heat release rate and geometry. It is assumed that these two effects are separable and that it is sufficient to deal with only the geometric dependency for this abstraction. Furthermore, since convection is predominant at the beginning of a fire, when the ceiling temperature rise is small, T surf T amb, the model is more concerned with making its most accurate prediction at that time. That is to say that, in the beginning of a fire, the radial temperature gradient across the ceiling surface is small enough to be ignored. So, since the ceiling surface temperature is essentially constant at this time, another assumption is made: that Eq (A)1 q = h * (T gas T surf ) can be replaced with Eq (A)2 q h * T gas Thus, the difference between these two formulations (Eq (A)1 and 2.3-4(A)2) is assumed to be a constant of proportionality. Also, this point selection process 32

44 only, the emphasis is on the trend of the heat flux with varying r/h, not necessarily the absolute magnitude of it. The local values of the heat flux will be calculated at the locations resulting from this abstraction process. Therefore, both h and T gas now become functions of r/h so that the total convection heat transfer to the ceiling proper is: Eq (A)3 q = (r/h) max 0 2 πh(t )dr gas which with a change of variable, r = (r/h)h, becomes: Eq (A)4 q /H 2 = (r/h) max 0 2πr/H)h(T gas )d(r/h) The functions for h and T gas are composed of expressions for the impingement zone (0 r/h < 0.2) and for the region outside the impingement zone (0.2 r/h (r/h) max). Therefore, the above integral is actually the sum of two integrals: Eq (A)5 q /H 0. 2 (r/h) max 2 = 2π(r/H)h(Tgas )d(r/h) π(r/h)h(t gas )d(r/h) As it turns out, the integral between 0 and 0.2 (first term of Eq (A)5) can be found analytically. Therefore, there is no 33

45 error associated with this region. A point at r/h = 0.0 and another at r/h = 0.2 are required to characterize the heat transfer in the impingement region. However, the expression for 0.2 < r/h < 2.2 (i.e., 2.2 is the limiting (r/h) max for ooper's expressions) is not so well behaved and requires a numerical technique. The numerical integration of the second term of Eq (A)5 was done with a Simpson's rule algorithm taken from Ref evenly spaced intervals were used to calculate the solution correct to four decimal places and thus this is considered to be the "correct' solution. The problem now becomes how to arrive at the "36 interval" solution using a finite number of not necessarily evenly spaced intervals and then where (i.e., at which values of r/h) to place the interval boundaries. Typically, a minimum of three points are required to span the entire space between r/h = 0 and (r/h) max : r/h = 0.0, 0.2 and (r/h) max. In other words, the emphasis is placed on the heat transfer associated with the impingement zone while accepting the error associated with outer region. So, for the region outside the impingement zone at least one point is required. If the error associated with the outer region is to be reduced, then more points are required. The maximum number of outer region intervals is arbitrarily set to six. In order to size the intervals appropriately, a variable interval generator is required. 34

46 The error associated with a variable interval generator is proportional to the fourth derivative (f iv ) of the function being integrated (Eq (A)5). This is comparable to the error associated with the Simpson s rule numerical integration subroutine provided by Ref. 3. For example, in Ref. 3, this error is shown to be: Eq (A)6 E S N iv f ( )(h/ 2 )(b a) =, a < < b 180 where: h = interval size a = lower bound b = upper bound This equation indicates that knowledge of the fourth derivative is required to estimate the error associated with the procedure described below. Therefore, by evaluating the fourth derivative, the order of magnitude of the error can be estimated. The required fourth derivative was also found numerically and the algorithm employed was taken from Ref. 14. Double precision was used and a smooth fourth derivative resulted, as shown in Fig (A)1. The fourth derivative is employed as follows: since the fourth derivative may represent the error associated with the interval h i. 35

47 36

48 One way to minimize the error is to find: min { f i iv * h i 4 ( )} (or, as shown by Eq (A)8, min { f i iv 1/4 * h i ( )}, subject to: Eq (A)7 where f i iv n i= 1 h( ) = (r/h) max 0. 2 is the "error" associated with h i ( ) at some r/h =. In order to do this, set Eq (A)8a iv 4 f i * h i = constant, K. By setting this product equal to a constant, the error associated with any interval is no greater than that for any other. This constant is actually a function of (r/h) max and the number of intervals, n: Eq (A)9 K = * r n where: = c(n, (r/h) max ) = function to account for the number of intervals, n, and the total area involved represented by (r/h) max : to be developed below r n = (r/h) max / n Now, from Eq (A)8a we have Eq (A)8b h i = K / (f I iv ) 1/4 The next step is to plug in for K and r n and then to sum over the intervals. This results in: 37

49 Eq (A)10 n i= 1 where: b - a (r/h) max Solving for results in: h i n n * rn *{(b a) / n} = = b a 1/ 4 = iv iv 1/ 4 i= 1 f i= 1 f i i Eq (A)11a Eq (A) 11b n = n / (1/ f i ) or n / (r / i= 1 H) max 0.2 (1/ f iv i iv 1/ 4 1/ 4 )d(r / H) Now plugging into the expression for a single interval, Eq (A)8b, h i = (r/h) results in: Eq (A)12a or (r / H) = *{[(r / H) f max iv 1/ 4 i 0.2]/ n} Eq (A)12b (r / H) {n / = (r / H) max 0.2 (1/ f 1/ 4 iv i )d(r f / H)}*{[(r / H) iv 1/ 4 i max 0.2]/ n} Before finding the interval sizes, and ultimately the points where the local conditions are to be calculated, values for iv i 1/ 4 f and iv i 1/ 4 1/ f are required as functions of r/h. These functional relationships are shown in Fig (A)2 and 2.3-4(A)3. The curve of Fig

50 4(A)2 asymptotically approaches x = 0. The spike shown in Fix 2.3-4(A)3 is caused by the function of Fig (A) crossing the X axis at approximately r/h = The data of Fig (A)2 were coded into a function subroutine that returns a value of iv 1/ 4 f i given r/h. Also, it is the integral of the data of Fig (A)3 between r/h = 0.2 and (r/h) max that is required. Therefore, the values of X, given by: Eq (A)13 X = (r / H) max 0.2 (1/ f iv 1/ 4 i )d(r / H) for a range of (r/h) max were found using a Simpson's rule algorithm and then tabulated in a DATA statement to provide values for the term in the numerator of Eq (A)12b. An algorithm was devised to iteratively find the local r/h's at the interval boundaries. The required input for this algorithm is the maximum value of r/h and the number of intervals (i.e., points) to be considered. The algorithm then tried to calculate a variable mesh by starting at r/h = 0.2 and ending at (r/h) max by "spanning the space" between these two points. A flowchart of this algorithm is shown in Fig (A)4. Unfortunately, this approach did not yield usable results. The fourth derivative is recognized to be important when determining the error, as shown by Eq

51 40

52 41

53 42

54 4(A)6. However, the fourth derivative used in this algorithm is not numerically well behaved. It appears that the upper bound on the problem, (r/h) max 2.2, in conjunction with the wide range of values for iv 1/ 4 f i, , is too restrictive to provide a rigorous solution. This algorithm would continually calculate interval sizes greater than r/h = 2.2 and/or consistently find the maximum value and nothing else. The problem seems to be numerical in nature. Therefore, a less rigorous approach will be used. It appears that one road leading to a spacing scheme involves some intuition. From the rigorous scheme detailed above, and from Fig (A)2, it seems that the region between r/h = 0.2 and 1.1 would be the region of greatest concern since the fourth derivative is largest there. In other words, the region where the error (i.e., the fourth derivative of the function in question) is largest is where the majority of the points should be located: large error, small intervals. Based on this observation, three spacing schemes were considered: (A) evenly spaced between r/h = 0.2 and (r/h) max; (B) evenly spaced between 0.2 and 1.1, with one point at (r/h) max (number of points in outer region 2 and (r/h) max > 1.1); and () evenly spaced between 0.2 and 1.1, one point at (r/h) max and another halfway between 1.1 and (r/h) max (number of points 3 and (r/h) max > 1.1). Each of these schemes (where applicable) is employed in 43

55 conjunction with six different values for (r/h) max as shown in Table 2.3-4(A)1. Since these "intuitive" schemes are assumed to be linear between successive points, a modified trapezoid rule was used to evaluate the integral between 0.2 and the various values of (r/h) max. The trapezoid rule was modified so that only two intervals between successive points are used: this implies that the functional relationship between adjacent values of h * T gas is linear. Thus, a number of estimates for the integral in question, second term of Eq (A)5, are obtained for the three different spacing schemes and six different values of (r/h) max. In addition, Simpson's rule was also used with the six values of (r/h) max to provide the basis for the comparison. The results of this comparison are also presented in Table 2.3-4(A)1. From this table it appears that an even spacing between 0.2 and (r/h) max is adequate for characterizing the convection heat transfer from the ceiling jet to the ceiling proper, for all values of (r/h) max. The "non-even" schemes tended to overpredict h * T gas : this is not conservative since too much heat would be convected. An even spacing doesn't appear to be unreasonable when a plot of (h * T gas ) vs. r/h, as shown in Fig (A)5, is seen. From this figure it can be seen that, although the function is complicated, its plot vs. r/h is not. 44

56 45

57 46

58 Upon closer examination of Fig (A)5, the effect of different spacing schemes (when used in conjunction with various values of (r/h) max) can readily be seen. An even spacing scheme should be the best fit since it tends to reduce the error associated with any given interval and thereby minimize the overall error. On the other hand, schemes B and result in one interval having an error relatively larger than the others in the scheme. For example, scheme with (r/h) max = 2.2 has points at 0.2, 0.35, 0.5, 0.65, 0.8, 1.65, 2.0 and 2.2. The interval between 0.8 and 1.65 has a much larger error associated with it because the curve (Fig (A)5) is concave up in that region (indeed, the curve is concave up between, roughly, 0.5 and 2.2). Therefore, the trapezoid rule over predicts in this region. This also explains the results presented in Table 2.3-4(A)1. Based on Fig (A)5, another spacing scheme that might reduce the overall error was considered. This scheme is unevenly spaced and linearly approximates the curve shown in Fig (A)5. The local points (values of r/h) are chosen as follows: in this scheme the number of points to be used depends on (r/h) max for the problem. That is to say 47

59 that the larger (r/h) max is, the more points that will be used. This will be accomplished by prescribing that certain points are always used, depending on (r/h) max. More explicitly, to minimize the error associated with assuming a linear relationship between adjacent points, the following points will always be used: 0.33, 0.53, 0.8, 1.1, 1.5, 1.9, and 2.2. If, for example, (r/h) max = 0.9, then four points outside the impingement zone are used: 0.33, 0.53, 0.8 and 0.9. Similarly, if (r/h) max = 1.8, then six points are used: 0.33, 0.53, 0.8, 1.1, 1.5, and 1.8. If (r/h) max > 2.2 then a total of eight points would be used: the seven specified above and (r/h) max. This should ensure that the "trapezoids" chosen fit the curve fairly well. A comparison of this latest scheme and Simpson's rule is provided in Table 2.3-4(A)2. Please note that the four place decimal accuracy shown in this table is based on T s = T amb and only of importance from a numerical standpoint. These results may not fully represent the accuracy of the more general case. Upon comparison with Table 2.3-4(A)1, the results of Table 2.3-4(A)2 are no worse, and in some cases better, than those of Table 2.3-4(A)2. Therefore, this uneven, predetermined spacing scheme is used in the alternate EHTX model. It provides acceptable results (all results within 1% of Simpson's rule) while speeding up the input processing. In other words, the argument is that this 48